theorem Th71:
  n <= m implies ATMOST+(B,n) c= ATMOST+(B,m)
  proof
    assume
A1: n <= m;
    let z be object; assume
A2: z in ATMOST+(B,n); then
    z in {y: card((y ref)\rng B) <= n} by Def35; then
    consider y such that
A3: z = y & card((y ref)\rng B) <= n;
    card((y ref)\rng B) <= m by A1,A3,XXREAL_0:2; then
    y in {x1 where x1 is Element of Y: card((x1 ref)\rng B) <= m};
    hence thesis by A2,A3,Def35;
  end;
